arXiv subjects

D. Thirumulanathan

Publications and source records attributed to D. Thirumulanathan.

3 recordsLinked to original sources

arXiv2019

Optimal Mechanisms for Selling Two Items to a Single Buyer Having Uniformly Distributed Valuations

We consider the design of a revenue-optimal mechanism when two items are available to be sold to a single buyer whose valuation is uniformly distributed over an arbitrary rectangle $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ in the positive quadrant. We provide an explicit, complete solution for arbitrary nonnegative values of $(c_1,c_2,b_1,b_2)$. We identify eight simple structures, each with at most $4$ (possibly stochastic) menu items, and prove that the optimal mechanism has one of these eight structures. We also characterize the optimal mechanism as a function of $(c_1,c_2,b_1,b_2)$. The structures indicate that the optimal mechanism involves (a) an interplay of individual sale and a bundle sale when $c_1$ and $c_2$ are low, (b) a bundle sale when $c_1$ and $c_2$ are high, and (c) an individual sale when one of them is high and the other is low. To the best of our knowledge, our results are the first to show the existence of optimal mechanisms with no exclusion region. We further conjecture, based on promising preliminary results, that our methodology can be extended to a wider class of distributions.

cs.GT↗

arXiv2019

On Optimal Mechanisms in the Two-Item Single-Buyer Unit-Demand Setting

We consider the problem of designing a revenue-optimal mechanism in the two-item, single-buyer, unit-demand setting when the buyer's valuations, $(z_1, z_2)$, are uniformly distributed in an arbitrary rectangle $[c,c+b_1]\times[c,c+b_2]$ in the positive quadrant. We provide a complete and explicit solution for arbitrary nonnegative values of $(c,b_1,b_2)$. We identify five simple structures, each with at most five (possibly stochastic) menu items, and prove that the optimal mechanism has one of the five structures. We also characterize the optimal mechanism as a function of $b_1, b_2$, and $c$. When $c$ is low, the optimal mechanism is a posted price mechanism with an exclusion region; when $c$ is high, it is a posted price mechanism without an exclusion region. Our results are the first to show the existence of optimal mechanisms with no exclusion region, to the best of our knowledge.

cs.GT↗

arXiv2017

Almost Budget Balanced Mechanisms with Scalar Bids For Allocation of a Divisible Good

This paper is about allocation of an infinitely divisible good to several rational and strategic agents. The allocation is done by a social planner who has limited information because the agents' valuation functions are taken to be private information known only to the respective agents. We allow only a scalar signal, called a bid, from each agent to the social planner. Yang and Hajek [Jour. on Selected Areas in Comm., 2007] as well as Johari and Tsitsiklis [Jour. of Oper. Res., 2009] proposed a scalar strategy Vickrey-Clarke-Groves (SSVCG) mechanism with efficient Nash equilibria. We consider a setting where the social planner desires minimal budget surplus. Example situations include fair sharing of Internet resources and auctioning of certain public goods where revenue maximization is not a consideration. Under the SSVCG framework, we propose a mechanism that is efficient and comes close to budget balance by returning much of the payments back to the agents in the form of rebates. We identify a design criterion for {\em almost budget balance}, impose feasibility and voluntary participation constraints, simplify the constraints, and arrive at a convex optimization problem to identify the parameters of the rebate functions. The convex optimization problem has a linear objective function and a continuum of linear constraints. We propose a solution method that involves a finite number of constraints, and identify the number of samples sufficient for a good approximation.

cs.GT↗